6533b856fe1ef96bd12b2641
RESEARCH PRODUCT
Darboux curves on surfaces I
Ronaldo GarciaRémi LangevinPaweł Walczaksubject
[ MATH ] Mathematics [math]GeodesicGeneral MathematicsDarboux frame02 engineering and technology01 natural sciencessymbols.namesakeMoving frame57R300202 electrical engineering electronic engineering information engineeringDarboux curves0101 mathematics[MATH]Mathematics [math]Möbius transformationMathematicsConformal geometryEuclidean spaceMSC: Primary 53A30 Secondary: 53C12 53C50 57R3053A3053C50010102 general mathematicsMathematical analysis53C12Ridge (differential geometry)Family of curvessymbolsSpace of spheres020201 artificial intelligence & image processingConformal geometrydescription
International audience; In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Mobius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable when the surface is a special canal.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2017-01-01 |